Three summation criteria for Fermat’s last theorem

Abstract

This paper extends the search for solutions of the congruences \[ ∑ 1 [ p / 6 ] 1 i ≡ 0 , ∑ 1 [ p / 6 ] 1 i 2 ≡ 0 and ∑ [ p / 6 ] + 1 [ p / 5 ] 1 i ≡ 0 ( mod p ) \sum \limits _1^{[p/6]} {\frac {1}{i} \equiv 0,} \quad \sum \limits _1^{[p/6]} {\frac {1}{{{i^2}}} \equiv 0} \quad {\text {and}}\quad \sum \limits _{[p/6] + 1}^{[p/5]} {\frac {1}{i} \equiv 0\;\pmod p} \] to the limit p > 600000 p > 600000 . The only solutions found were p = 61 p = 61 in the first case, in the second p = 205129 p = 205129 , and in the third case p = 109 p = 109 and p = 491 p = 491 .